Significance Instability is the Achilles’ heel of modern artificial intelligence (AI) and a paradox, with training algorithms finding unstable neural networks (NNs) despite existence stable ones. This foundational issue relates to Smale’s 18th mathematical problem for 21st century on limits AI. By expanding methodologies initiated by Gödel Turing, we demonstrate limitations (even randomized) computing NNs. Despite numerous results NNs great approximation properties, only in specific cases do...

Dynamic Mode Decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics turbulence and closely related Koopman operators. However, verifying decomposition, equivalently computed spectral features operators, remains major challenge due infinite-dimensional nature Challenges include spurious (unphysical) modes, dealing with continuous spectra, both which occur in turbulent flows....

Abstract Koopman operators are infinite‐dimensional that globally linearize nonlinear dynamical systems, making their spectral information valuable for understanding dynamics. However, can have continuous spectra and invariant subspaces, computing a considerable challenge. This paper describes data‐driven algorithms with rigorous convergence guarantees of from trajectory data. We introduce residual dynamic mode decomposition (ResDMD), which provides the first scheme pseudospectra general...

.Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis decomposition of systems. However, Koopman are infinite dimensional, computing considerable challenge. We introduce measure-preserving extended dynamic mode (mpEDMD), first Galerkin method whose eigendecomposition converges to quantities general mpEDMD data-driven algorithm based on an orthogonal Procrustes problem that enforces truncations using dictionary...

Computing the spectra of operators is a fundamental problem in sciences, with wide-ranging applications condensed-matter physics, quantum mechanics and chemistry, statistical mechanics, etc. While there are algorithms that certain cases converge to spectrum, no general procedure known (a) always converges, (b) provides bounds on errors approximation, (c) approximate eigenvectors. This may lead incorrect simulations. It has been an open since 1950s decide whether such reliable methods exist...

Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The can achieve arbitrarily high orders convergence in terms a smoothing parameter general differential, integral, and lattice Explicit pointwise $L^p$-error bounds are derived local regularity measure. We provide numerical examples, including partial differential operator magnetic tight-binding model graphene, compute 1000 eigenvalues Dirac...

Abstract Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, many gaining approximation spectrum not enough. Often it vital to determine geometric features such as Lebesgue measure, capacity or fractal dimensions, different types spectral radii and numerical ranges, detect gaps essential corresponding failure finite section method. Despite new results on computing substantial interest these problems, there...

Passive-scalar mixing (metals, molecules, etc.) in the turbulent interstellar medium (ISM) is critical for abundance patterns of stars and clusters, galaxy star formation, cooling from circumgalactic medium. However, fundamental scaling laws remain poorly understood highly supersonic, magnetized, shearing regime relevant ISM. We therefore study full governing passive-scalar transport idealized simulations supersonic turbulence. Using simple phenomenological arguments variation diffusivity...

This paper establishes some of the fundamental barriers in theory computations and finally settles long-standing computational spectral problem. That is to determine existence algorithms that can compute spectra $\mathrm{sp}(A)$ classes bounded operators $A = \{a_{ij}\}_{i,j \in \mathbb{N}} \mathcal{B}(l^2(\mathbb{N}))$, given matrix elements $\{a_{ij}\}_{i,j \mathbb{N}}$, are sharp sense they achieve boundary what a digital computer achieve. Similarly, for Schrödinger operator $H -Δ+V$,...

Abstract

Abstract Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonance phenomena, and fluid stability analysis. Similarly, spectral decompositions (into pure point, absolutely continuous singular parts) often characterise relevant physical properties the long-time dynamics of systems. Despite new results on computing spectra, there remains no general method able to compute or infinite-dimensional normal operators. Previous efforts have focused...

We develop a rapid and accurate contour method for the solution of time-fractional PDEs. The inverts Laplace transform via an optimised stable quadrature rule, suitable infinite-dimensional operators, whose error decreases like $\exp(-cN/\log(N))$ $N$ points. is parallisable, avoids having to resolve singularities as $t\downarrow 0$, large memory consumption that can be challenge time-stepping methods applied ODEs resulting from are solved using adaptive sparse spectral converge...

The problem of computing spectra operators is arguably one the most investigated areas computational mathematics. However, general bounded infinite matrices has only recently been solved. We establish some foundations spectral theory through Solvability Complexity Index (SCI) hierarchy, an approach closely related to Smale’s program on mathematics and McMullen’s results polynomial root finding with rational maps. Infinite-dimensional problems yield intricate classification theory,...

Abstract Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these properties, and dynamic mode decomposition (DMD) stands out as the poster child projection-based methods. Although operator itself is linear, fact that it acts in an infinite-dimensional space observables poses challenges. These include spurious modes, essential spectra, verification decompositions. While recent...

This paper implements the unified transform to problems in unbounded domains with solutions having corner singularities. Consequently, a wide variety of mixed boundary condition can be solved without need for Wiener-Hopf technique. Such arise frequently acoustic scattering or calculation electric fields geometries involving finite and/or multiple plates. The new approach constructs global relation that relates known data, such as scattered normal velocity on rigid plate, unknown values, jump...

Recent work has given rise to a novel and simple numerical technique for solving elliptic boundary value problems formulated in convex polygons two dimensions. The method, based on the unified transform, involves expanding unknown values Legendre basis determining expansion coefficients by evaluating so-called global relation at appropriate points complex Fourier plane (spectral collocation). In this paper we provide significant advancement of providing fast efficient method evaluate...

We develop an algorithm that computes strongly continuous semigroups on infinite-dimensional Hilbert spaces with explicit error control. Given a generator $A$, time $t>0$, arbitrary initial vector $u_0$, and tolerance $\epsilon>0$, the $\exp(tA)u_0$ bounded by $\epsilon$. The is based combination of regularized functional calculus, suitable contour quadrature rules, adaptive computation resolvents in infinite dimensions. As particular case, we show it possible, even when only allowing...

We introduce the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm, which computes generalized eigenfunction decompositions of Koopman operators. By considering evolution observables, operators transform complex nonlinear dynamics into a linear framework suitable for spectral analysis. While powerful, traditional (DMD) techniques often struggle with continuous spectra. DMD addresses these challenges data-driven methodology that approximates operator's resolvent and its eigenfunctions...